G7 Government Yields

Initialize environment

library(caim)
caim::clr()

Setup preferences

# most important maturities
maturities_a = c(.25, 2, 5, 10)
# interesting maturities, level b
maturities_b = c(.0833, 1, 30)
# all other maturities will be level c

# relative weightings for each maturity level
# these will form a density function that will be normalised to sum to 1
#   , so don't worry about that here
MAT_A_WT = 1
MAT_B_WT = .75
MAT_C_WT = 0.15

# set of maturities to use for lambda values
lambda_maturities <- seq(0.5, 5, 0.5)

# k for k-fold evaluation
NUM_K_FOLDS = 10

Load G7 yield info

g7_rates_info <- fread("../../data/ratesinfo.csv")[
  country %in% c("US", "CA", "DE", "FR", "IT", "UK", "JP") & class=="GOVT"
  , .(country, class, maturity, id=bb_ticker)
]

g7_rates_data <- g7_rates_info[
  fread("../../data/ratesdata.csv")
  , 
  , on=.(id), nomatch=NULL 
][
  , .(country
      , class
      , maturity
      , date=lubridate::as_date(date)
      , year=lubridate::year(lubridate::as_date(date))
      , month=lubridate::month(lubridate::as_date(date))
      , yield=PX_LAST
      )
][
  , .SD, key=.(country, class, maturity, date)
]


g7_curve_mats <- sort(unique(g7_rates_data$maturity))
g7_curve_columns <- paste0("yield_", g7_curve_mats)

g7_curves <- dcast(
  g7_rates_data, country + class + date + year + month ~ maturity, value.var="yield"  
)[
  , .SD
  , key=.(country, class, date)  
]
setnames(g7_curves, c(key(g7_curves), g7_curve_mats), c(key(g7_curves), g7_curve_columns))

g7_curves[]

Load G7 Money market data

g7_mm_info <- data.table(
  country=c("US", "CA", "DE", "FR", "IT", "UK", "JP", "DE", "FR", "IT")
  , curncy=c("USD", "CAD", "EUR", "EUR", "EUR", "GBP", "JPY", "DEM", "FRF", "ITL")
  , id=c("USD.FIX.1M", "CAD.FIX.1M", "EUR.FIX.1M", "EUR.FIX.1M", "EUR.FIX.1M"
         , "GBP.FIX.1M", "JPY.FIX.1M", "DEM.FIX.1M", "FRF.FIX.1M", "ITL.FIX.1M")
)
# g7_mm_info <- fread("../data/mminfo.csv")[
#   CURNCY %in% c("USD",  "EUR", "GBP", "JPY", "CAD")
#   , .(curncy=CURNCY, id=FIX_1M)
# ]
g7_mm_data <- g7_mm_info[
  fread("../../data/mmdata.csv")
  , 
  , on=.(id), nomatch=NULL
][
  , .(country
      , curncy
      , date = lubridate::as_date(date)
      , year=lubridate::year(lubridate::as_date(date))
      , month=lubridate::month(lubridate::as_date(date))
      , maturity = 0.8333
      , yield = PX_LAST
      )
][
  , .SD , key=.(country, curncy, date)
]

# stitch together pre- and post-EUR convergence data for DE, FR, IT
eur_start_date <- min(g7_mm_data[country=="DE" & curncy=="EUR"]$date)
# delete local observations where EUR data is available
g7_mm_data <- g7_mm_data[!(curncy %in% c("DEM", "FRF", "ITL") & date >= eur_start_date)]
# relabel pre-EUR local observations as EUR
# may need to rethink this if we want to implement pre- and post-EUR currency data
# - maybe label everything as DEM FRF ITL?
g7_mm_data[curncy %in% c("DEM", "FRF", "ITL"), curncy := "EUR"] 

g7_mm_data[]

Create monthly data

g7_curves_m <- g7_curves[
  date %in% caim::month_end_dates(date)
]

g7_mm_data_m <- g7_mm_data[
  date %in% caim::month_end_dates(date)
]

Add money market data to short end of government curves

This is, admittedly, a big kludge, but should help when there are no government yields below 2-year maturity, like early Canada, etc. Also, this may be a good way to average 3-month bills and 1-month deposits as a proxy for cash when doing Nelson-Siegel analyses.

Assumption: We’ll do LIBOR - 1/8 as a crude proxy for LIBID

g7_curves_m[g7_mm_data_m, yield_0.0833 := i.yield - 0.125, on=.(country, year, month)][]

Ensure there are at least 3 yield points

# ensure there are at least 3 yield points
g7_curves_m <- g7_curves_m[
  g7_curves_m[ ,.(valid=(sum(!is.na(.SD)) >= 3)), by=.(country, class, date)]$valid
][
  # find first date that works for all countries
  date >= max(g7_curves_m[,.(min_date=min(date)), by=.(country, class)]$min_date)
]

Calculate Nelson Siegel coefficients

curve_data <- g7_curves_m
curve_mats <- g7_curve_mats
curve_columns <- g7_curve_columns

curve_coefs <- list()
curve_ids <- unique(curve_data[,.(country, class)])
for (c in 1:nrow(curve_ids)) {
  t_curve_data <- curve_data[country == curve_ids[c, country] & class == curve_ids[c, class]]
  t_curve_dates <- sort(unique(t_curve_data$date))
  t_curve_mats <- curve_mats #sort(unique(yield_info[curve_id==c]$maturity))
  t_curve_mat_names <- curve_columns #as.character(t_curve_mats)
  
  t_curve_mat_wts <- rep(MAT_C_WT, length(t_curve_mats))
  names(t_curve_mat_wts) <- t_curve_mat_names
  t_curve_mat_wts[names(t_curve_mat_wts) %in% maturities_a] <- MAT_A_WT
  t_curve_mat_wts[names(t_curve_mat_wts) %in% maturities_b] <- MAT_B_WT
  t_curve_mat_wts <- t_curve_mat_wts / sum(t_curve_mat_wts)
  
  
  t_res <- list()
  for (i in 1:length(lambda_maturities)) {
    mat <- lambda_maturities[i]
    lambda <- caim::ns_mat2lambda(mat)
    print(paste("calculating for curve:", c, curve_ids[c, country], curve_ids[c, class], "maturity:", mat, "lambda:", lambda))
    t_res[[i]] <- list(
      mat=mat
      , lambda=lambda
      , coefs=data.table(
        country=curve_ids[c, country]
        , class=curve_ids[c, class]
        , date=t_curve_data[,date]
        , t(apply(t_curve_data, 1, function(x) 
          caim::ns_yields2coefs(t_curve_mats
                                , as.numeric(x[t_curve_mat_names])
                                , lambda=lambda
                                , wts=t_curve_mat_wts)))
        , lambda_mat=mat
        )
      )
  }
  
  # calculate summaries
  # 5 year halflife assuming 260 business days in a year
  decay <- halflife2decay(5*260)
  lambda_summary <- data.table(t(sapply(t_res, function(x) 
    c(mat=x$mat, lambda=x$lambda, wss=mean(x$coefs$wss), wss_exp=mean_exp(x$coefs$wss, decay)))))

  # choose best data
  best_hist_fit <- which(lambda_summary$wss == min(lambda_summary$wss))
  best_exp_fit <- which(lambda_summary$wss_exp == min(lambda_summary$wss_exp))
  # take average of best fits, but tilt towards best_exp_fit
  best_ix <- floor(mean(c(best_hist_fit, best_exp_fit))+ifelse(best_exp_fit > best_hist_fit, 0.5, 0))
  best_hist_coefs <- t_res[[best_ix]]$coefs
  # best_hist_coefs$curve_id <- c

  #add data to curve_coefs
  curve_coefs[[length(curve_coefs)+1]] <- best_hist_coefs
}

[1] "calculating for curve: 1 CA GOVT maturity: 0.5 lambda: 3.58657038690834"
[1] "calculating for curve: 1 CA GOVT maturity: 1 lambda: 1.79328063484169"
[1] "calculating for curve: 1 CA GOVT maturity: 1.5 lambda: 1.19552075559691"
[1] "calculating for curve: 1 CA GOVT maturity: 2 lambda: 0.896641550467829"
[1] "calculating for curve: 1 CA GOVT maturity: 2.5 lambda: 0.717323931054624"
[1] "calculating for curve: 1 CA GOVT maturity: 3 lambda: 0.59776033052309"
[1] "calculating for curve: 1 CA GOVT maturity: 3.5 lambda: 0.512348203420652"
[1] "calculating for curve: 1 CA GOVT maturity: 4 lambda: 0.448335176139679"
[1] "calculating for curve: 1 CA GOVT maturity: 4.5 lambda: 0.398496343358628"
[1] "calculating for curve: 1 CA GOVT maturity: 5 lambda: 0.358681854090179"
[1] "calculating for curve: 2 DE GOVT maturity: 0.5 lambda: 3.58657038690834"
[1] "calculating for curve: 2 DE GOVT maturity: 1 lambda: 1.79328063484169"
[1] "calculating for curve: 2 DE GOVT maturity: 1.5 lambda: 1.19552075559691"
[1] "calculating for curve: 2 DE GOVT maturity: 2 lambda: 0.896641550467829"
[1] "calculating for curve: 2 DE GOVT maturity: 2.5 lambda: 0.717323931054624"
[1] "calculating for curve: 2 DE GOVT maturity: 3 lambda: 0.59776033052309"
[1] "calculating for curve: 2 DE GOVT maturity: 3.5 lambda: 0.512348203420652"
[1] "calculating for curve: 2 DE GOVT maturity: 4 lambda: 0.448335176139679"
[1] "calculating for curve: 2 DE GOVT maturity: 4.5 lambda: 0.398496343358628"
[1] "calculating for curve: 2 DE GOVT maturity: 5 lambda: 0.358681854090179"
[1] "calculating for curve: 3 FR GOVT maturity: 0.5 lambda: 3.58657038690834"
[1] "calculating for curve: 3 FR GOVT maturity: 1 lambda: 1.79328063484169"
[1] "calculating for curve: 3 FR GOVT maturity: 1.5 lambda: 1.19552075559691"
[1] "calculating for curve: 3 FR GOVT maturity: 2 lambda: 0.896641550467829"
[1] "calculating for curve: 3 FR GOVT maturity: 2.5 lambda: 0.717323931054624"
[1] "calculating for curve: 3 FR GOVT maturity: 3 lambda: 0.59776033052309"
[1] "calculating for curve: 3 FR GOVT maturity: 3.5 lambda: 0.512348203420652"
[1] "calculating for curve: 3 FR GOVT maturity: 4 lambda: 0.448335176139679"
[1] "calculating for curve: 3 FR GOVT maturity: 4.5 lambda: 0.398496343358628"
[1] "calculating for curve: 3 FR GOVT maturity: 5 lambda: 0.358681854090179"
[1] "calculating for curve: 4 IT GOVT maturity: 0.5 lambda: 3.58657038690834"
[1] "calculating for curve: 4 IT GOVT maturity: 1 lambda: 1.79328063484169"
[1] "calculating for curve: 4 IT GOVT maturity: 1.5 lambda: 1.19552075559691"
[1] "calculating for curve: 4 IT GOVT maturity: 2 lambda: 0.896641550467829"
[1] "calculating for curve: 4 IT GOVT maturity: 2.5 lambda: 0.717323931054624"
[1] "calculating for curve: 4 IT GOVT maturity: 3 lambda: 0.59776033052309"
[1] "calculating for curve: 4 IT GOVT maturity: 3.5 lambda: 0.512348203420652"
[1] "calculating for curve: 4 IT GOVT maturity: 4 lambda: 0.448335176139679"
[1] "calculating for curve: 4 IT GOVT maturity: 4.5 lambda: 0.398496343358628"
[1] "calculating for curve: 4 IT GOVT maturity: 5 lambda: 0.358681854090179"
[1] "calculating for curve: 5 JP GOVT maturity: 0.5 lambda: 3.58657038690834"
[1] "calculating for curve: 5 JP GOVT maturity: 1 lambda: 1.79328063484169"
[1] "calculating for curve: 5 JP GOVT maturity: 1.5 lambda: 1.19552075559691"
[1] "calculating for curve: 5 JP GOVT maturity: 2 lambda: 0.896641550467829"
[1] "calculating for curve: 5 JP GOVT maturity: 2.5 lambda: 0.717323931054624"
[1] "calculating for curve: 5 JP GOVT maturity: 3 lambda: 0.59776033052309"
[1] "calculating for curve: 5 JP GOVT maturity: 3.5 lambda: 0.512348203420652"
[1] "calculating for curve: 5 JP GOVT maturity: 4 lambda: 0.448335176139679"
[1] "calculating for curve: 5 JP GOVT maturity: 4.5 lambda: 0.398496343358628"
[1] "calculating for curve: 5 JP GOVT maturity: 5 lambda: 0.358681854090179"
[1] "calculating for curve: 6 UK GOVT maturity: 0.5 lambda: 3.58657038690834"
[1] "calculating for curve: 6 UK GOVT maturity: 1 lambda: 1.79328063484169"
[1] "calculating for curve: 6 UK GOVT maturity: 1.5 lambda: 1.19552075559691"
[1] "calculating for curve: 6 UK GOVT maturity: 2 lambda: 0.896641550467829"
[1] "calculating for curve: 6 UK GOVT maturity: 2.5 lambda: 0.717323931054624"
[1] "calculating for curve: 6 UK GOVT maturity: 3 lambda: 0.59776033052309"
[1] "calculating for curve: 6 UK GOVT maturity: 3.5 lambda: 0.512348203420652"
[1] "calculating for curve: 6 UK GOVT maturity: 4 lambda: 0.448335176139679"
[1] "calculating for curve: 6 UK GOVT maturity: 4.5 lambda: 0.398496343358628"
[1] "calculating for curve: 6 UK GOVT maturity: 5 lambda: 0.358681854090179"
[1] "calculating for curve: 7 US GOVT maturity: 0.5 lambda: 3.58657038690834"
[1] "calculating for curve: 7 US GOVT maturity: 1 lambda: 1.79328063484169"
[1] "calculating for curve: 7 US GOVT maturity: 1.5 lambda: 1.19552075559691"
[1] "calculating for curve: 7 US GOVT maturity: 2 lambda: 0.896641550467829"
[1] "calculating for curve: 7 US GOVT maturity: 2.5 lambda: 0.717323931054624"
[1] "calculating for curve: 7 US GOVT maturity: 3 lambda: 0.59776033052309"
[1] "calculating for curve: 7 US GOVT maturity: 3.5 lambda: 0.512348203420652"
[1] "calculating for curve: 7 US GOVT maturity: 4 lambda: 0.448335176139679"
[1] "calculating for curve: 7 US GOVT maturity: 4.5 lambda: 0.398496343358628"
[1] "calculating for curve: 7 US GOVT maturity: 5 lambda: 0.358681854090179"

ns_coefs <- rbindlist(curve_coefs)[
  , .(ns_beta0 = beta0
      , ns_beta1 = beta1
      , ns_beta2 = beta2
      , ns_lambda = lambda
      , ns_lambda_mat = lambda_mat
      , ns_wss = wss
      )
  , key=.(country, class, date)
]

g7_curves_m <- g7_curves_m[ns_coefs]

rm(list=setdiff(ls(), c("g7_curves", "g7_curves_m", "g7_rates_data", "g7_rates_info", "g7_curve_columns", "g7_curve_mats")))

Yield calculations

long_data <- melt(
  g7_curves_m  
  , c("country", "class", "date", "ns_beta0", "ns_beta1", "ns_beta2", "ns_lambda")
  ,  patterns("yield_")
  , value.name="yield_now"
)[
  , maturity := as.numeric(stringr::str_remove(variable, "yield_"))
][
  , .(country, class, date, ns_beta0, ns_beta1, ns_beta2, ns_lambda, maturity, yield_now)
]

# fill in missing yields with interpolated data
# table with !is.na(yield)
yields_valid <- long_data[!is.na(yield_now)][, c("mat", "yld") := .(maturity, yield_now)]
# table mapping to <=
yields_lo <- yields_valid[long_data, .(country, date, maturity, m_lo0=mat, y_lo0=yld), on=.(country, date, maturity), roll=T]
# table mapping to >=
yields_hi <- yields_valid[long_data, .(country, date, maturity, m_hi0=mat, y_hi0=yld), on=.(country, date, maturity), roll=-Inf]

yields_lin <- yields_hi[
  yields_lo
  , .(
    country
    , date
    , maturity
    , yield_lin = ifelse(m_hi0 == m_lo0
                         , y_hi0
                         , y_lo0 + (maturity - m_lo0) * (y_hi0 - y_lo0) / (m_hi0 - m_lo0)
                         )
    , y_hi0
    , y_lo0
    , m_hi0
    , m_lo0
  )
  , on=.(country, date, maturity)
]

long_data <- long_data[yields_lin, on=.(country, date, maturity)]

long_data <- long_data[
  , c("num_coups", "mod_dur") := .(
    ifelse(maturity > 0.25, ifelse(country=="US", 2, 1), 0)
    , maturity
  )  
][
  maturity > 0.25
  , mod_dur := round(caim::modified_duration(yield_lin / 100, yield_lin / 100, maturity, num_coups), 6)
][
  , .(
    ns_beta0
    , ns_beta1
    , ns_beta2
    , ns_lambda
    , num_coups
    , mod_dur #= round(caim::modified_duration(yield_now / 100, yield_now / 100, maturity, 1), 6)
    , yield_now
    , yield_lin
    , y_hi0
    , y_lo0
    , m_hi0
    , m_lo0
    )
  , key=.(country, class, maturity, date)
][
  , yield_prev := shift(yield_lin), by = .(country, class, maturity)
][
  , c("y_hi1", "y_lo1", "m_hi1", "m_lo1") := .(
    yield_lin
    , shift(yield_lin)
    , maturity
    , shift(maturity)
  )
  , by = .(country, class, date)
][
  , yield_sell_lin := (
    y_lo1 + ((m_hi1 - 1/12) - m_lo1) * (y_hi1 - y_lo1) / (m_hi1 - m_lo1)
  )
][
  , yield_buy_ns := round(caim::ns_coefs2yields(
    mats = maturity
    , beta0 = shift(ns_beta0)
    , beta1 = shift(ns_beta1)
    , beta2 = shift(ns_beta2)
    , lambda = shift(ns_lambda)
  )$y, 4)
  , by = .(country, class, maturity)
][
  , yield_sell_ns := round(caim::ns_coefs2yields(
    mats = maturity - 1/2
    , beta0 = ns_beta0
    , beta1 = ns_beta1
    , beta2 = ns_beta2
    , lambda = ns_lambda
  )$y, 4)
][
  , coup_inc_lin := round(yield_prev / 1200, 6)
][
  , shift_inc := ifelse(maturity > 0.25, round((yield_lin - yield_prev) / 100 * -mod_dur, 6), 0)
][
  , tot_ret_shift := coup_inc_lin + shift_inc
][
  , dur_inc_lin := ifelse(
    maturity > 0.25
    , round(caim::bond_price(yield_sell_lin / 100, yield_prev / 100, maturity - 1/12, 1, 1) - 1, 6)
    , 0
    )
][
  , tot_ret_lin := coup_inc_lin + dur_inc_lin
][
  , coup_inc_ns := round(yield_buy_ns / 1200, 6)  
][
 , price_inc_ns := ifelse(
   maturity > 0.25
   , round(caim::bond_price(yield_sell_ns / 100, yield_buy_ns / 100, maturity - 1/12, 1, 1) - 1, 6)
   , 0
   )
][
  , tot_ret_ns := coup_inc_ns + price_inc_ns
]

long_data[country=="US"]

wide_data <- dcast(
  long_data
  ,country + class + date + ns_beta0 + ns_beta1 + ns_beta2 + ns_lambda ~ maturity
  , value.var = names(long_data)[!(names(long_data) %in% c("country", "class", "date", "ns_beta0", "ns_beta1", "ns_beta2", "ns_lambda", "maturity"))]

  # , value.var = c("mod_dur", "yield_now", "yield_prev", "yield_buy_ns", "yield_sell_ns"
  #                 , "coup_inc_lin", "dur_inc_lin", "coup_inc_ns", "price_inc_ns", "tot_ret_ns"
  #                 )
  )

wide_data[]

g7_return_data <- dcast(
  long_data[maturity %in% c(0.0833, 1, 2, 3, 5, 7, 10)]
  , country + class + date ~ maturity
  , value.var = c("coup_inc_lin", "tot_ret_shift", "tot_ret_lin", "tot_ret_ns")
)

g7_return_data[]

Get asset data

g7_assets <- data.table(
  id=c("US.GOVT.13", "US.GOVT.15", "US.GOVT.110"
       , "CA.GOVT.13", "CA.GOVT.15", "CA.GOVT.110"
       , "DE.GOVT.13", "DE.GOVT.15", "DE.GOVT.110"
       , "FR.GOVT.13", "FR.GOVT.15", "FR.GOVT.110"
       , "IT.GOVT.13", "IT.GOVT.15", "IT.GOVT.110"
       , "UK.GOVT.13", "UK.GOVT.15", "UK.GOVT.110"
       , "JP.GOVT.13", "JP.GOVT.15", "JP.GOVT.110"
       )
  , country=c(rep("US", 3), rep("CA", 3), rep("DE", 3), rep("FR", 3), rep("IT", 3)
              , rep("UK", 3), rep("JP", 3))
  , maturity=rep(c(13, 15, 110), 7)
  , key=c("country", "maturity")
)
g7_asset_info <- fread("../../data/assetinfo.csv")[suggname %in% g7_assets$id]

g7_asset_info[]

g7_asset_data <- g7_assets[
  fread("../../data/assetdata.csv")
  , .(
    country
    , maturity
    , id
    , date = lubridate::as_date(date)
    , year = lubridate::year(date)
    , month = lubridate::month(date)
    , index_nav=PX_LAST
    )
  , on=.(id), nomatch=NULL
][
  date %in% caim::month_end_dates(date)
  , .SD
  , key=.(country, maturity, date)
][
  , index_ret := round(index_nav / shift(index_nav) - 1, 6)
  , by=.(country, maturity)
]

g7_asset_data[]

Model Setup

g7_models <- list(
  list(name="x_g13_coup_bullet", type="coup_bullet", maturity=13
       , features=c("coup_inc_lin_2"), unity_coefs=T)
  , list(name="x_g13_shift_bullet", type="shift_bullet", maturity=13
         , features=c("tot_ret_shift_2"), unity_coefs=T)
  , list(name="x_g13_lin_bullet", type="lin_bullet", maturity=13
         , features=c("tot_ret_lin_2"), unity_coefs=T)
  , list(name="x_g13_ns_bullet", type="ns_bullet", maturity=13
         , features=c("tot_ret_ns_2"), unity_coefs=T)
  , list(name="x_g13_coup_n", type="coup_n", maturity=13
         , features=c("coup_inc_lin_1", "coup_inc_lin_2", "coup_inc_lin_3"), unity_coefs=T)
  , list(name="x_g13_shift_n", type="shift_n", maturity=13
         , features=c("tot_ret_shift_1", "tot_ret_shift_2", "tot_ret_shift_3"), unity_coefs=T)
  , list(name="x_g13_lin_n", type="lin_n", maturity=13
         , features=c("tot_ret_lin_1", "tot_ret_lin_2", "tot_ret_lin_3"), unity_coefs=T)
  , list(name="x_g13_ns_n", type="ns_n", maturity=13
         , features=c("tot_ret_ns_1", "tot_ret_ns_2", "tot_ret_ns_3"), unity_coefs=T)
  , list(name="x_g13_coup_ladder", type="coup_ladder", maturity=13
         , features=c("coup_inc_lin_1", "coup_inc_lin_2", "coup_inc_lin_3"), unity_coefs=F)
  , list(name="x_g13_shift_ladder", type="shift_ladder", maturity=13
         , features=c("tot_ret_shift_1", "tot_ret_shift_2", "tot_ret_shift_3"), unity_coefs=F)
  , list(name="x_g13_lin_ladder", type="lin_ladder", maturity=13
         , features=c("tot_ret_lin_1", "tot_ret_lin_2", "tot_ret_lin_3"), unity_coefs=F)
  , list(name="x_g13_ns_ladder", type="ns_ladder", maturity=13
         , features=c("tot_ret_ns_1", "tot_ret_ns_2", "tot_ret_ns_3"), unity_coefs=F)

  , list(name="x_g15_coup_bullet", type="coup_bullet", maturity=15
         , features=c("coup_inc_lin_3"), unity_coefs=T)
  , list(name="x_g15_shift_bullet", type="shift_bullet", maturity=15
         , features=c("tot_ret_shift_3"), unity_coefs=T)
  , list(name="x_g15_lin_bullet", type="lin_bullet", maturity=15
         , features=c("tot_ret_lin_3"), unity_coefs=T)
  , list(name="x_g15_ns_bullet", type="ns_bullet", maturity=15
         , features=c("tot_ret_ns_3"), unity_coefs=T)
  , list(name="x_g15_coup_n", type="coup_n", maturity=15
         , features=c("coup_inc_lin_1", "coup_inc_lin_2", "coup_inc_lin_3", "coup_inc_lin_5"), unity_coefs=T)
  , list(name="x_g15_shift_n", type="shift_n", maturity=15
         , features=c("tot_ret_shift_1", "tot_ret_shift_2", "tot_ret_shift_3", "tot_ret_shift_5"), unity_coefs=T)
  , list(name="x_g15_lin_n", type="lin_n", maturity=15
         , features=c("tot_ret_lin_1", "tot_ret_lin_2", "tot_ret_lin_3", "tot_ret_lin_5"), unity_coefs=T)
  , list(name="x_g15_ns_n", type="ns_n", maturity=15
         , features=c("tot_ret_ns_1", "tot_ret_ns_2", "tot_ret_ns_3", "tot_ret_ns_5"), unity_coefs=T)
  , list(name="x_g15_coup_ladder", type="coup_ladder", maturity=15
         , features=c("coup_inc_lin_1", "coup_inc_lin_2", "coup_inc_lin_3", "coup_inc_lin_5"), unity_coefs=F)
  , list(name="x_g15_shift_ladder", type="shift_ladder", maturity=15
         , features=c("tot_ret_shift_1", "tot_ret_shift_2", "tot_ret_shift_3", "tot_ret_shift_5"), unity_coefs=F)
  , list(name="x_g15_lin_ladder", type="lin_ladder", maturity=15
         , features=c("tot_ret_lin_1", "tot_ret_lin_2", "tot_ret_lin_3", "tot_ret_lin_5"), unity_coefs=F)
  , list(name="x_g15_ns_ladder", type="ns_ladder", maturity=15
         , features=c("tot_ret_ns_1", "tot_ret_ns_2", "tot_ret_ns_3", "tot_ret_ns_5"), unity_coefs=F)
  
  , list(name="x_g110_coup_bullet", type="coup_bullet", maturity=110
         , features=c("coup_inc_lin_5"), unity_coefs=T)
  , list(name="x_g110_shift_bullet", type="shift_bullet", maturity=110
         , features=c("tot_ret_shift_5"), unity_coefs=T)
  , list(name="x_g110_lin_bullet", type="lin_bullet", maturity=110
         , features=c("tot_ret_lin_5"), unity_coefs=T)
  , list(name="x_g110_ns_bullet", type="ns_bullet", maturity=110
         , features=c("tot_ret_ns_5"), unity_coefs=T)
  , list(name="x_g110_coup_n", type="coup_n", maturity=110
         , features=c("coup_inc_lin_1", "coup_inc_lin_2", "coup_inc_lin_3", "coup_inc_lin_5", "coup_inc_lin_7", "coup_inc_lin_10")
         , unity_coefs=T)
  , list(name="x_g110_shift_n", type="shift_n", maturity=110
         , features=c("tot_ret_shift_1", "tot_ret_shift_2", "tot_ret_shift_3", "tot_ret_shift_5", "tot_ret_shift_7", "tot_ret_shift_10")
         , unity_coefs=T)
  , list(name="x_g110_lin_n", type="lin_n", maturity=110
         , features=c("tot_ret_lin_1", "tot_ret_lin_2", "tot_ret_lin_3", "tot_ret_lin_5", "tot_ret_lin_7", "tot_ret_lin_10")
         , unity_coefs=T)
  , list(name="x_g110_ns_n", type="ns_n", maturity=110
         , features=c("tot_ret_ns_1", "tot_ret_ns_2", "tot_ret_ns_3", "tot_ret_ns_5", "tot_ret_ns_7", "tot_ret_ns_10")
         , unity_coefs=T)
  , list(name="x_g110_coup_ladder", type="coup_ladder", maturity=110
         , features=c("coup_inc_lin_1", "coup_inc_lin_2", "coup_inc_lin_3", "coup_inc_lin_5", "coup_inc_lin_7", "coup_inc_lin_10")
         , unity_coefs=F)
  , list(name="x_g110_shift_ladder", type="shift_ladder", maturity=110
         , features=c("tot_ret_shift_1", "tot_ret_shift_2", "tot_ret_shift_3", "tot_ret_shift_5", "tot_ret_shift_7", "tot_ret_shift_10")
         , unity_coefs=F)
  , list(name="x_g110_lin_ladder", type="lin_ladder", maturity=110
         , features=c("tot_ret_lin_1", "tot_ret_lin_2", "tot_ret_lin_3", "tot_ret_lin_5", "tot_ret_lin_7", "tot_ret_lin_10")
         , unity_coefs=F)
  , list(name="x_g110_ns_ladder", type="ns_ladder", maturity=110
         , features=c("tot_ret_ns_1", "tot_ret_ns_2", "tot_ret_ns_3", "tot_ret_ns_5", "tot_ret_ns_7", "tot_ret_ns_10")
         , unity_coefs=F)
)

g7_features <- sort(unique(rbindlist(lapply(g7_models, function(x) data.table(feature=x$features))))$feature)

Model evaluation - full history

K-fold cross validation

k_fold_eval <- rbindlist(apply(g7_assets, 1, function(asset) {
  y_data <- g7_asset_data[id==asset["id"]]
  x_data <- g7_return_data[country==asset["country"]][
    , c("year", "month") := .(
      lubridate::year(date)
      , lubridate::month(date)
    )
  ]
  # print(paste(asset["country"], asset["maturity"], nrow(y_data), nrow(x_data)))
  model_data <- y_data[
    x_data
    ,
    , on=.(year, month), nomatch=NULL
  ][
    , !c(
      "country", "i.country", "class", "maturity", "year", "month", "index_nav", "i.date"
    )
  ]

  has_invalid <- rowSums(is.na(model_data[ , ..g7_features])) > 0
  # print(paste(asset["id"], "rows:", nrow(model_data), "invalid:", sum(has_invalid)))

  model_data <- model_data[!has_invalid]
  # print(paste("clean rows:", nrow(model_data)))
  
  k_ix <- caim::k_fold_ix(nrow(model_data), NUM_K_FOLDS)
  
  asset_models <- lapply(which(sapply(g7_models
                                      , function(x) x$maturity==as.numeric(asset["maturity"])))
                         , function(x) {
                           g7_models[[x]] 
                         })
  
  model_res <- lapply(asset_models, function (m) {
    print(paste(asset["id"], m$name))
    y <- model_data[ , index_ret]
    X <- cbind(intercept=1, model_data[ , m$features, with=F])
    
    k_res <- sapply(seq(1, NUM_K_FOLDS), function(fold) {
      train_ix <- k_ix[k != fold, ix]
      test_ix <- k_ix[k == fold, ix]
      
      train_y <- y[train_ix]
      train_X <- X[train_ix]

      Aeq <- c(0, rep(1, ncol(train_X) - 1))
      beq <- 1
      lb <- c(caim::NEG_INF, rep(0, ncol(train_X) - 1))
      wts <- caim::d_exp(caim::halflife2decay(12), nrow(train_X))
  
      fixed_coefs <- NULL
      if (m$unity_coefs)
        fixed_coefs <- c(0, rep(1/length(m$features), length(m$features)))
      
      lmq <- caim::lm_quad(train_y, train_X
                           , Aeq=Aeq
                           , beq=beq
                           , lb=lb
                           , wts=wts
                           , fixed_coefs=fixed_coefs
                           )
      
      test_y <- y[test_ix]
      test_X <- X[test_ix]
      
      # print(test_X)
      test_y_hat <- as.matrix(test_X) %*% lmq$coefs
      residuals <- test_y_hat - test_y
      R2 <- cor(test_y, test_y_hat) ^ 2
      RMSE <- sqrt(mean(residuals^2))
      alpha_y <- (1 + mean(residuals)) ^ 12 - 1
      te_y <- sd(residuals) * sqrt(12)

      return(round(c(k=fold, R2=R2, RMSE=RMSE, alpha_y=alpha_y, te_y=te_y), 6))

    })
    
    return(data.table(asset=asset["id"], model=m$name, type=m$type, t(k_res)))

  })

  return(rbindlist(model_res))

}))[
  , .(
    R2_mean = mean(R2)
    , RMSE_mean = mean(RMSE)
    , alpha_y_mean = mean(alpha_y)
    , te_y_mean = mean(te_y)
    , R2_sd = sd(R2)
    , RMSE_sd = sd(RMSE)
    , alpha_y_sd = sd(alpha_y)
    , te_y_sd = sd(te_y)
    , score = mean(RMSE) * sd(RMSE)
  )
  , by=.(asset, model, type)
  ][
    g7_assets, on=c(asset="id")
  ][
    , .SD
    , key=.(country, maturity, score)
  ]

[1] "CA.GOVT.13 x_g13_coup_bullet"
[1] "CA.GOVT.13 x_g13_shift_bullet"
[1] "CA.GOVT.13 x_g13_lin_bullet"
[1] "CA.GOVT.13 x_g13_ns_bullet"
[1] "CA.GOVT.13 x_g13_coup_n"
[1] "CA.GOVT.13 x_g13_shift_n"
[1] "CA.GOVT.13 x_g13_lin_n"
[1] "CA.GOVT.13 x_g13_ns_n"
[1] "CA.GOVT.13 x_g13_coup_ladder"
[1] "CA.GOVT.13 x_g13_shift_ladder"
[1] "CA.GOVT.13 x_g13_lin_ladder"
[1] "CA.GOVT.13 x_g13_ns_ladder"
[1] "CA.GOVT.15 x_g15_coup_bullet"
[1] "CA.GOVT.15 x_g15_shift_bullet"
[1] "CA.GOVT.15 x_g15_lin_bullet"
[1] "CA.GOVT.15 x_g15_ns_bullet"
[1] "CA.GOVT.15 x_g15_coup_n"
[1] "CA.GOVT.15 x_g15_shift_n"
[1] "CA.GOVT.15 x_g15_lin_n"
[1] "CA.GOVT.15 x_g15_ns_n"
[1] "CA.GOVT.15 x_g15_coup_ladder"
[1] "CA.GOVT.15 x_g15_shift_ladder"
[1] "CA.GOVT.15 x_g15_lin_ladder"
[1] "CA.GOVT.15 x_g15_ns_ladder"
[1] "CA.GOVT.110 x_g110_coup_bullet"
[1] "CA.GOVT.110 x_g110_shift_bullet"
[1] "CA.GOVT.110 x_g110_lin_bullet"
[1] "CA.GOVT.110 x_g110_ns_bullet"
[1] "CA.GOVT.110 x_g110_coup_n"
[1] "CA.GOVT.110 x_g110_shift_n"
[1] "CA.GOVT.110 x_g110_lin_n"
[1] "CA.GOVT.110 x_g110_ns_n"
[1] "CA.GOVT.110 x_g110_coup_ladder"
[1] "CA.GOVT.110 x_g110_shift_ladder"
[1] "CA.GOVT.110 x_g110_lin_ladder"
[1] "CA.GOVT.110 x_g110_ns_ladder"
[1] "DE.GOVT.13 x_g13_coup_bullet"
[1] "DE.GOVT.13 x_g13_shift_bullet"
[1] "DE.GOVT.13 x_g13_lin_bullet"
[1] "DE.GOVT.13 x_g13_ns_bullet"
[1] "DE.GOVT.13 x_g13_coup_n"
[1] "DE.GOVT.13 x_g13_shift_n"
[1] "DE.GOVT.13 x_g13_lin_n"
[1] "DE.GOVT.13 x_g13_ns_n"
[1] "DE.GOVT.13 x_g13_coup_ladder"
[1] "DE.GOVT.13 x_g13_shift_ladder"
[1] "DE.GOVT.13 x_g13_lin_ladder"
[1] "DE.GOVT.13 x_g13_ns_ladder"
[1] "DE.GOVT.15 x_g15_coup_bullet"
[1] "DE.GOVT.15 x_g15_shift_bullet"
[1] "DE.GOVT.15 x_g15_lin_bullet"
[1] "DE.GOVT.15 x_g15_ns_bullet"
[1] "DE.GOVT.15 x_g15_coup_n"
[1] "DE.GOVT.15 x_g15_shift_n"
[1] "DE.GOVT.15 x_g15_lin_n"
[1] "DE.GOVT.15 x_g15_ns_n"
[1] "DE.GOVT.15 x_g15_coup_ladder"
[1] "DE.GOVT.15 x_g15_shift_ladder"
[1] "DE.GOVT.15 x_g15_lin_ladder"
[1] "DE.GOVT.15 x_g15_ns_ladder"
[1] "DE.GOVT.110 x_g110_coup_bullet"
[1] "DE.GOVT.110 x_g110_shift_bullet"
[1] "DE.GOVT.110 x_g110_lin_bullet"
[1] "DE.GOVT.110 x_g110_ns_bullet"
[1] "DE.GOVT.110 x_g110_coup_n"
[1] "DE.GOVT.110 x_g110_shift_n"
[1] "DE.GOVT.110 x_g110_lin_n"
[1] "DE.GOVT.110 x_g110_ns_n"
[1] "DE.GOVT.110 x_g110_coup_ladder"
[1] "DE.GOVT.110 x_g110_shift_ladder"
[1] "DE.GOVT.110 x_g110_lin_ladder"
[1] "DE.GOVT.110 x_g110_ns_ladder"
[1] "FR.GOVT.13 x_g13_coup_bullet"
[1] "FR.GOVT.13 x_g13_shift_bullet"
[1] "FR.GOVT.13 x_g13_lin_bullet"
[1] "FR.GOVT.13 x_g13_ns_bullet"
[1] "FR.GOVT.13 x_g13_coup_n"
[1] "FR.GOVT.13 x_g13_shift_n"
[1] "FR.GOVT.13 x_g13_lin_n"
[1] "FR.GOVT.13 x_g13_ns_n"
[1] "FR.GOVT.13 x_g13_coup_ladder"
[1] "FR.GOVT.13 x_g13_shift_ladder"
[1] "FR.GOVT.13 x_g13_lin_ladder"
[1] "FR.GOVT.13 x_g13_ns_ladder"
[1] "FR.GOVT.15 x_g15_coup_bullet"
[1] "FR.GOVT.15 x_g15_shift_bullet"
[1] "FR.GOVT.15 x_g15_lin_bullet"
[1] "FR.GOVT.15 x_g15_ns_bullet"
[1] "FR.GOVT.15 x_g15_coup_n"
[1] "FR.GOVT.15 x_g15_shift_n"
[1] "FR.GOVT.15 x_g15_lin_n"
[1] "FR.GOVT.15 x_g15_ns_n"
[1] "FR.GOVT.15 x_g15_coup_ladder"
[1] "FR.GOVT.15 x_g15_shift_ladder"
[1] "FR.GOVT.15 x_g15_lin_ladder"
[1] "FR.GOVT.15 x_g15_ns_ladder"
[1] "FR.GOVT.110 x_g110_coup_bullet"
[1] "FR.GOVT.110 x_g110_shift_bullet"
[1] "FR.GOVT.110 x_g110_lin_bullet"
[1] "FR.GOVT.110 x_g110_ns_bullet"
[1] "FR.GOVT.110 x_g110_coup_n"
[1] "FR.GOVT.110 x_g110_shift_n"
[1] "FR.GOVT.110 x_g110_lin_n"
[1] "FR.GOVT.110 x_g110_ns_n"
[1] "FR.GOVT.110 x_g110_coup_ladder"
[1] "FR.GOVT.110 x_g110_shift_ladder"
[1] "FR.GOVT.110 x_g110_lin_ladder"
[1] "FR.GOVT.110 x_g110_ns_ladder"
[1] "IT.GOVT.13 x_g13_coup_bullet"
[1] "IT.GOVT.13 x_g13_shift_bullet"
[1] "IT.GOVT.13 x_g13_lin_bullet"
[1] "IT.GOVT.13 x_g13_ns_bullet"
[1] "IT.GOVT.13 x_g13_coup_n"
[1] "IT.GOVT.13 x_g13_shift_n"
[1] "IT.GOVT.13 x_g13_lin_n"
[1] "IT.GOVT.13 x_g13_ns_n"
[1] "IT.GOVT.13 x_g13_coup_ladder"
[1] "IT.GOVT.13 x_g13_shift_ladder"
[1] "IT.GOVT.13 x_g13_lin_ladder"
[1] "IT.GOVT.13 x_g13_ns_ladder"
[1] "IT.GOVT.15 x_g15_coup_bullet"
[1] "IT.GOVT.15 x_g15_shift_bullet"
[1] "IT.GOVT.15 x_g15_lin_bullet"
[1] "IT.GOVT.15 x_g15_ns_bullet"
[1] "IT.GOVT.15 x_g15_coup_n"
[1] "IT.GOVT.15 x_g15_shift_n"
[1] "IT.GOVT.15 x_g15_lin_n"
[1] "IT.GOVT.15 x_g15_ns_n"
[1] "IT.GOVT.15 x_g15_coup_ladder"
[1] "IT.GOVT.15 x_g15_shift_ladder"
[1] "IT.GOVT.15 x_g15_lin_ladder"
[1] "IT.GOVT.15 x_g15_ns_ladder"
[1] "IT.GOVT.110 x_g110_coup_bullet"
[1] "IT.GOVT.110 x_g110_shift_bullet"
[1] "IT.GOVT.110 x_g110_lin_bullet"
[1] "IT.GOVT.110 x_g110_ns_bullet"
[1] "IT.GOVT.110 x_g110_coup_n"
[1] "IT.GOVT.110 x_g110_shift_n"
[1] "IT.GOVT.110 x_g110_lin_n"
[1] "IT.GOVT.110 x_g110_ns_n"
[1] "IT.GOVT.110 x_g110_coup_ladder"
[1] "IT.GOVT.110 x_g110_shift_ladder"
[1] "IT.GOVT.110 x_g110_lin_ladder"
[1] "IT.GOVT.110 x_g110_ns_ladder"
[1] "JP.GOVT.13 x_g13_coup_bullet"
[1] "JP.GOVT.13 x_g13_shift_bullet"
[1] "JP.GOVT.13 x_g13_lin_bullet"
[1] "JP.GOVT.13 x_g13_ns_bullet"
[1] "JP.GOVT.13 x_g13_coup_n"
[1] "JP.GOVT.13 x_g13_shift_n"
[1] "JP.GOVT.13 x_g13_lin_n"
[1] "JP.GOVT.13 x_g13_ns_n"
[1] "JP.GOVT.13 x_g13_coup_ladder"
[1] "JP.GOVT.13 x_g13_shift_ladder"
[1] "JP.GOVT.13 x_g13_lin_ladder"
[1] "JP.GOVT.13 x_g13_ns_ladder"
[1] "JP.GOVT.15 x_g15_coup_bullet"
[1] "JP.GOVT.15 x_g15_shift_bullet"
[1] "JP.GOVT.15 x_g15_lin_bullet"
[1] "JP.GOVT.15 x_g15_ns_bullet"
[1] "JP.GOVT.15 x_g15_coup_n"
[1] "JP.GOVT.15 x_g15_shift_n"
[1] "JP.GOVT.15 x_g15_lin_n"
[1] "JP.GOVT.15 x_g15_ns_n"
[1] "JP.GOVT.15 x_g15_coup_ladder"
[1] "JP.GOVT.15 x_g15_shift_ladder"
[1] "JP.GOVT.15 x_g15_lin_ladder"
[1] "JP.GOVT.15 x_g15_ns_ladder"
[1] "JP.GOVT.110 x_g110_coup_bullet"
[1] "JP.GOVT.110 x_g110_shift_bullet"
[1] "JP.GOVT.110 x_g110_lin_bullet"
[1] "JP.GOVT.110 x_g110_ns_bullet"
[1] "JP.GOVT.110 x_g110_coup_n"
[1] "JP.GOVT.110 x_g110_shift_n"
[1] "JP.GOVT.110 x_g110_lin_n"
[1] "JP.GOVT.110 x_g110_ns_n"
[1] "JP.GOVT.110 x_g110_coup_ladder"
[1] "JP.GOVT.110 x_g110_shift_ladder"
[1] "JP.GOVT.110 x_g110_lin_ladder"
[1] "JP.GOVT.110 x_g110_ns_ladder"
[1] "UK.GOVT.13 x_g13_coup_bullet"
[1] "UK.GOVT.13 x_g13_shift_bullet"
[1] "UK.GOVT.13 x_g13_lin_bullet"
[1] "UK.GOVT.13 x_g13_ns_bullet"
[1] "UK.GOVT.13 x_g13_coup_n"
[1] "UK.GOVT.13 x_g13_shift_n"
[1] "UK.GOVT.13 x_g13_lin_n"
[1] "UK.GOVT.13 x_g13_ns_n"
[1] "UK.GOVT.13 x_g13_coup_ladder"
[1] "UK.GOVT.13 x_g13_shift_ladder"
[1] "UK.GOVT.13 x_g13_lin_ladder"
[1] "UK.GOVT.13 x_g13_ns_ladder"
[1] "UK.GOVT.15 x_g15_coup_bullet"
[1] "UK.GOVT.15 x_g15_shift_bullet"
[1] "UK.GOVT.15 x_g15_lin_bullet"
[1] "UK.GOVT.15 x_g15_ns_bullet"
[1] "UK.GOVT.15 x_g15_coup_n"
[1] "UK.GOVT.15 x_g15_shift_n"
[1] "UK.GOVT.15 x_g15_lin_n"
[1] "UK.GOVT.15 x_g15_ns_n"
[1] "UK.GOVT.15 x_g15_coup_ladder"
[1] "UK.GOVT.15 x_g15_shift_ladder"
[1] "UK.GOVT.15 x_g15_lin_ladder"
[1] "UK.GOVT.15 x_g15_ns_ladder"
[1] "UK.GOVT.110 x_g110_coup_bullet"
[1] "UK.GOVT.110 x_g110_shift_bullet"
[1] "UK.GOVT.110 x_g110_lin_bullet"
[1] "UK.GOVT.110 x_g110_ns_bullet"
[1] "UK.GOVT.110 x_g110_coup_n"
[1] "UK.GOVT.110 x_g110_shift_n"
[1] "UK.GOVT.110 x_g110_lin_n"
[1] "UK.GOVT.110 x_g110_ns_n"
[1] "UK.GOVT.110 x_g110_coup_ladder"
[1] "UK.GOVT.110 x_g110_shift_ladder"
[1] "UK.GOVT.110 x_g110_lin_ladder"
[1] "UK.GOVT.110 x_g110_ns_ladder"
[1] "US.GOVT.13 x_g13_coup_bullet"
[1] "US.GOVT.13 x_g13_shift_bullet"
[1] "US.GOVT.13 x_g13_lin_bullet"
[1] "US.GOVT.13 x_g13_ns_bullet"
[1] "US.GOVT.13 x_g13_coup_n"
[1] "US.GOVT.13 x_g13_shift_n"
[1] "US.GOVT.13 x_g13_lin_n"
[1] "US.GOVT.13 x_g13_ns_n"
[1] "US.GOVT.13 x_g13_coup_ladder"
[1] "US.GOVT.13 x_g13_shift_ladder"
[1] "US.GOVT.13 x_g13_lin_ladder"
[1] "US.GOVT.13 x_g13_ns_ladder"
[1] "US.GOVT.15 x_g15_coup_bullet"
[1] "US.GOVT.15 x_g15_shift_bullet"
[1] "US.GOVT.15 x_g15_lin_bullet"
[1] "US.GOVT.15 x_g15_ns_bullet"
[1] "US.GOVT.15 x_g15_coup_n"
[1] "US.GOVT.15 x_g15_shift_n"
[1] "US.GOVT.15 x_g15_lin_n"
[1] "US.GOVT.15 x_g15_ns_n"
[1] "US.GOVT.15 x_g15_coup_ladder"
[1] "US.GOVT.15 x_g15_shift_ladder"
[1] "US.GOVT.15 x_g15_lin_ladder"
[1] "US.GOVT.15 x_g15_ns_ladder"
[1] "US.GOVT.110 x_g110_coup_bullet"
[1] "US.GOVT.110 x_g110_shift_bullet"
[1] "US.GOVT.110 x_g110_lin_bullet"
[1] "US.GOVT.110 x_g110_ns_bullet"
[1] "US.GOVT.110 x_g110_coup_n"
[1] "US.GOVT.110 x_g110_shift_n"
[1] "US.GOVT.110 x_g110_lin_n"
[1] "US.GOVT.110 x_g110_ns_n"
[1] "US.GOVT.110 x_g110_coup_ladder"
[1] "US.GOVT.110 x_g110_shift_ladder"
[1] "US.GOVT.110 x_g110_lin_ladder"
[1] "US.GOVT.110 x_g110_ns_ladder"

k_fold_best_models <- k_fold_eval[
  , head(.SD, 3)
  , by=.(country, maturity, asset)
  ]

best_model_freq <- sort(table(k_fold_best_models$type), decreasing=T)
best_model_freq

       lin_n   lin_ladder    ns_ladder shift_ladder      shift_n   lin_bullet    ns_bullet         ns_n 
          18           16            9            8            7            3            1            1 

Old appendix: constrained linear least squares = quadratic programming

t(sapply(names(g13_models), function(x) {
  model <- g13_models[[x]]
  y <- model_data[ , index_ret]
  X <- cbind(intercept=1, model_data[ , model$features, with=F])

  Aeq <- c(0, rep(1, ncol(X) - 1))
  beq <- 1
  lb <- c(caim::NEG_INF, rep(0, ncol(X) - 1))
  wts <- caim::d_exp(caim::halflife2decay(12), nrow(X))
  
  fixed_coefs <- NULL
  if (model$unity_coefs)
    fixed_coefs <- c(0, rep(1/length(model$features), length(model$features)))

  lmq <- caim::lm_quad(y, X, Aeq=Aeq, beq=beq, lb=lb, wts=wts, fixed_coefs=fixed_coefs)
  # coefs <- round(lmq$coefs, 6)
  # print(paste(x, coefs))
  
  print(lmq$coefs)
  return(round(evaluate(lmq), 6))
  # return(as.matrix(c(model=x, t(round(evaluate(lmq), 6)))))
}))